UvA-DARE (Digital Academic Repository)
Beat-to-beat blood-pressure fluctuations and heart-rate variability in man:
physiological relationships, analysis techniques and a simple model
de Boer, R.W.
Link to publication
Citation for published version (APA):
de Boer, R. W. (1985). Beat-to-beat blood-pressure fluctuations and heart-rate variability in man: physiological
relationships, analysis techniques and a simple model
General rights
It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s),
other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).
Disclaimer/Complaints regulations
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating
your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask
the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam,
The Netherlands. You will be contacted as soon as possible.
UvA-DARE is a service provided by the library of the University of Amsterdam (http://dare.uva.nl)
Download date: 16 Jun 2017
Chapter 7
Relationships between short-term blood-pressure fluctuations
and heart-rate variability:
II.
A simple model
In this chapter we give a tentative interpretation of the phase spectra of
RR-intervals against pressure variables (chapter 6) by means of the beat-to-beat
model that was presented in chapter 5.
7.1 Abstract
A simple model of the beat-to-beat properties of the cardiovascular system is
used to interpret the results of spectral analysis of blood-pressure and interval
data as presented in chapter 6. The model consists of two equations, one representing the fast regulation of interval by the systolic pressure (baroreflex),
the other one representing a Windkessel approximation of the systemic arterial
system.
The model, When applied to interval and blood-pressure data from resting subjects, explains the lack of respiratory variability in the diastolic pressure values. The baroreflex equation seems to describe the data only in the region of
respiratory frequencies. The shape of the phase spectrum of systolic pressures
against intervals is modelled by difference equations, but no physiological interpretation of these equations is given.
7.2 Introduction
In chapter 6 we presented power spectra of beat-to-beat RR-interval (RRI) and
blood pressure (BP) fluctuations.
These spectra showed that both RRI-and
BP-variability can be attributed quantitatively to its various causes, particularly
to respiratory influences and the IO-second-rhythm.It was remarkable that almost no respiratory influence was seen in the spectrum of diastolic pressures.
We also presented cross-spectra of blood pressure values against intervals, which
showed that systolic pressure variations lead interval variations by about two
beats (2 s) in the lO-second-region; however, in the respiratory region, the intervals seemed to vary together with the systolic pressures.
105
In the present paper we try to explain some of the results and indicate how
the cross-spectra may be interpreted.
Our point of departure is a simple
beat-to-beat model of the cardiovascular system (chapter 5, or DeBoer et aI.,
1983).
7.3 Methods
Blood pressure and ECG from resting subjects were recorded. Beat-to-beat
pressure values and intervals were derived from these recordings and were used
to calculate power and cross-spectra. Successi ve interval and pressure values
were considered to be equidistantly spaced at distances equal to the mean intervallength T. Details of the data acquisition and the estimation of the spectra are given in chapter 6.
The cross-spectrum between a pressure variable (e.g., systolic pressure S or diastolic pressure D) and the interval I consists of two parts.
The (squared)
coherence spectrum k 2(f) is a measure of the correlation between pressure and
interval variability in a certain frequency band, and has values between 0 (no
correlation) and 1 (complete correlation). The phase spectrum
<p(f)
indicates
the phase difference between the signals; a negative value of <p(f) implies the
pressure variation to lead the interval variation. If the coherence is low, the
phase cannot be estimated reliably (Jenkins and Watts, 1968, ch.9).
In some cases, the data were band-pass filtered by applying a Digital Fourier
Transform to the vai.IJes, setting the unwanted contributions equal to zero, and
'applying an inverse Fourier Transform.
7.4 Model
7.4.1 Relations between beat-to-beat blood-pressure and R-R interval values.
To interpret the power and cross-spectra, we start with part of the simple
beat-to-beat model as presented in a previous chapter (chapter 5). The following, physiologically plausible relationships between successive pressure and interval values were used to describe experimental BP- and RRI-data:
a) In
b) Dn
::
a o. Sn +
C1
c2,Sn_l.exp(-In_11 T
(baroref lex)
(Windkessel)
In these equations Sn and Dn are the systolic and diastolic pressure, respectively, occurring during RR-interval In'
106
Eqn.a) states that an increase in systolic pressure Sn implies an immediate increase in interval-length In' This is due to the fast, vagally mediated baroreflex-loop, aimed at keeping the blood presSure constant (Sleight, 1980). When a
systolic pressure wave arrives at the baroreceptors in the aorta and the carotid
sinuses, it causes a burst of afferent nerve spikes, travelling towards the Central Nervous System (CNS). The amount of nerve output depends on the pressure felt by the baroreceptors (Arndt et al., 1977). The pressure information is
processed in the CNS, which then sends an efferent spike burst along the vagal
nerve towards the cardiac pacemaker, thus leading to an adjustable amount of
delay of subsequent heart beats. (The sympathetic effects on heart rate are
neglected here, as they are much slower.) The most simple description of this
process is eqn.a). As we consider data from resting subjects with small variations around the mean values, a linear approximation for the relationship
between Sn and In is acceptable.
The constant a o is the so-called baroreflex-sensitivity coefficient and has values
in the range 5-30 ms/mmHg (Smyth et aI., 1969; Pickering et ai., 1972).
These values were found in experiments in which the blood pressure was increased or decreased by drugs, and the subsequent change in interval, was compared with the change in blood pressure.
Eqn.b) is a formulation of the Windkessel model for the systemic circulation
during diastolic run-off. The value of the diastolic pressure Dn depends on the
previous systolic pressure Sn-l and on the length of the previous interval In_I'
i
is the time-constant of the Windkessel. The equation can be linearized by
rearranging and taking logarithms:
b') Ln
d~f. log(Sn_l/Dn)
==
In-II T - log(c2),
with Ln the logarithmic decrement.
In chapter 5 eqns.a) and b') were found to be in reasonable agreement with experimental BP- and RRI-data. In the next sections the phase spectra as implied
by these equations are compared with spectra as calculated from the experimental data of fig.2 of chapter 6. The mean interval length I of the 960 considered beats was 0.93 s. The power spectra of intervals (fig.la), of systolic
pressures (fig.lb, solid line) and of diastolic pressures (fig.lb, dashed) are again
107
0.0150
a
140.0
1-
P (tl
2
5 21HZ
b
P (fl
(mmHg)
5-
.
1Hz .
0- - --
0.0075
c
180
1. a
0-1
90
o
180
d
tl
I
I
I
I
<p
\
\
\
1\
II 1\
III I
I
o. 5H--iI~....;I~~--+--lt+--+.~I
I
/
-90
(f)
90
I
I
I "I
o
I I
I I
'\
-90
I
o. a
0.0
\ I
.f
"=-,-=~---:c'-:-~-=-'- 180
0.5
frequency (Hz)
Fig.7-1a Power spectrum P(f) of RR-intervals from a resting subject,
showing peaks at 0.1 Hz (1 O-second-rhythm) and around the respiratory
frequency.
Fig.7-1,b Power spectrum P(f) oisystolicpressure S (solid line) and of
diastq1ic pressures D (dashed line). In the latter spectrum the respiratory peak is absent.
Fig.7-lc Squared coherence spectrum k2(f) (dashed line, between 0 and
1), and phase spectrum q> (f) (solid line, between -180 0 and 1800 ) of
systolic pressure S against interval I. A high coherence implies a
strong link between pressure and interval variations, as is the case around 0.1 Hz and in the respiratory region. When the coherence is
high (> 0.5), the phase is reliably estimated (heavy line). The phase
is negative when pressure variations lead interval variations (e.g., at
0.1 Hz). In the region of respiratory frequencies, the phase difference
between pressure and interval variations is small.
Fig.7-1d Squared coherence spectrum k2(f) (dashed line) and phase
spectrum q> (f) (solid line) of diastolic pressure D against interval I.
The phase spectrum shows no trend.
108
presented here, as are the cross-spectra of systolic and diastolic pressure against RR-interval (figs.lc and Id, respectively). The dashed lines in figs.lc,d are
the (squared) coherence spectra k 2(f), between 0 and 1, and the solid lines are
the phase spectra, with values between -180 0 and 180 0 •
7.4.2 The baroreflex equation
The baroreflex equation a) implies that no phase difference exists between variations in S and in I. Fig.lc shows that this phase difference is only approximately zero in the region of respiratory frequency (0.20-0.35 Hz).
0.1 Hz a definite lead of pressure exists of around 60 0
= 1/6
Around
th period, cor-
responding with 10/6 = 1.7 s or approximately 2 beats. Hence eqn.a) does not
describe the data over the whole frequency range from 0 to 0.5 Hz. The
scatter plot of In against Sn (fig.2a), which shows no clear linear relationship
(correlation coefficient r = 0.25), does not corroborate equation a) either.
Thus, the phase spectrum between I and S (Fig. Ie) suggests that the
baroreflex-equation a) is valid only in the region of normal respiratory frequency. An estimate of the baroreflex sensitivity coefficient a o should therefore
be based on this frequency-band only. In order to study only res,piratory variations, the following simple method was used. The 0.1 Hz waves and other
low-frequency fluctuations were removed from the BP- and RRI-variations by
differencing of successive pressure or interval values, i.e.,
xg = Xn - Xn_l'
This is equivalent to filtering the data with a high-pass filter with gain-function
4.sin 2(1T fi) (fig.2b; Jenkins and Watts', 1968, Ch.7.3).
a
1200
interval I
n
(ms)
1100
1000
0'
.m
D
0. 0 '
0 .....
41"0 0
\ii~o
.
900
800
700~
120
..
Fig.7-2a Scatter plot of R-R intervals
In against systolic pressures Sn occurring
during this interval. The baroreflex
equation implies a linear relationship
between In and Sn' which is not seen
from the data.
____~____~~____~____~
140
160
systol i c pressure Sn (mmHg)
109
4.0
b
gain
Fig.7 -2b Gain-function of the difference
filter as used to remove the
low-frequency variability from I and S.
As the mean interval length Yis 0.93 s,
the filter is defined from 0 to 1/2T =
0.54 Hz.
2.0
c
0.0200
140.0
P (f)
P (f)
5 21HZ
d
(mmHg) 21Hz
70.0
0.0100
o.ooooL~-L~L-..L----L-----.l_..L:::±:::::I:::=
0.0
0.1
0.2
0.3
0.4
0.5
frequency (Hz)
frequency
Fig.7-2c Power spectrum of RR-intervals after application of the
difference fil ter to remove the variability due to the 10-second-rhythm
and to slower fluctuations. Almost only respiratory variability is left.
Fig.7-2d Power spectrum of systolic pressures after application of the
difference filter.
e
1.0
/ __ /
r
I
I
!
I
/ . ,.......\
180
cp
V\
\
(f)
90
\
\
\
Fig.7-2e Squared coherence spectrum
o k2(f) (dashed) and phase spectrum
-90
Ll.!----L------L_.L--.L--.l_-"-------'--_L----L---l-180
0.1
0.2
0.3
0.4
0.5
frequency (Hz)
110
<p(f)
(solid) of differenced intervals against
differenced systolic pressures. No appreciable difference with fig.lc. See text.
differenced
interval rd
100
50
... .
Fig.7-2f Scatter plot of differenced intervals
= In-1n_1 against differenced
systolic
pressures
= 5o-Sn-I' These
• di f ferenced
systolic d differenced values appear to be linearly
r;:~9iure sn related (d. fig.2a).
-50
-100
•
Ig
sg
~
o
-10
10
For the data of fig. 1a,b this technique produces power spectra of Id and of Sd
as shown in Figs.2c,d.
cross-spectrum of
The 0.1 Hz peaks are now minimized.
sci against
The
Id is not changed (fig.2e), because the phase shift
due to the difference filter is identical for ~ and for Sd. The existing minor
differences between fig. Ie and fig.2e are due to the averaging of the spectral
values, preceding the calculation of the cross-spectrum. Fig.2f is the scatter
plot of
Ig against sg.
in fig.2a
(r
As expected, the correlation coefficient is higher than
= 0.76),
permitting an estimation
of
the
coefficient
a o = 9.7 ms/mmHg.
A different estimate of the barorefJex sensitivity coefficient was obtained by
applying a band-pass fil terbetween 0.2 Hz and 0.35 Hz to the data (see
section 7.2); linear regression of the filtered data of intervals on systolic pressures now lead to a value a o = 13.2 ms/mmHg (r = 0.94). This value, which
lies within the wide range of values obtained by other authors (section 7.4.1),
will be used later on.
7.4.3 The Windkessel equation
According to the linearized equation b') a fixed delay of one beat exists
between the logarithmic decrement L and I; this amounts to a phase difference between L and I increasing linearly with frequency from 00 at f = 0 to
1800 for f
= 1/2i = 0.54
Hz.
This is actually observed in the calculated
cross-spectrum of L against I (fig.3a).
III
1.0
Fig.7-3a Squared coherence spectrum
k2(f) (dashed line) and phase spectrum
(') \
o.5H~--If------+--r----ti o cp (f) (solid line) of logarithmic decrement
\" I
\/
L <Ln = log(Sn-l/Dn» against interval I.
I -90 The approximately linear increase of the
phase validates the
"7"---'--=-,-:--''--:::-'-;:;--L~-;:;--'--;:;-,-;~-;;-,-1BO Windkessel-equation b ' ) (see text).
0.5
90
I'
frequency (Hz)
We estimated the timeconstant
r
by linear regression of Ln on In-I' after re-
moving frequency components under 0.05 Hz from I and L. This was done because for these frequencies the power in the spectra of I and L is high, but
the coherence is low. Hence the correlation between Ln and In_l improves if
the low-frequency component is removed. After filtering the time constant was
r=
estimated as
1.85 s (r
=0.87),
which is in agreement with published va-
lues for the time constant of a Windkessel approximation to the systemic circulation (Simon et al., 1979;
Watt and Burrus, 1976).
The evidence presented above suggests that the baroreflex equation a) holds
only in the region of the normal respiratory frequency (0.20-0.35 Hz), but that
the Windkessel equation b ') is approximately valid for all frequencies above
0.05 Hz. In the following, egns.a) and b ' ) are used to explain why a respiratory
peak should be absent in the spectrum of diastolic pressures.
We assume small variations of D n, Sn and In around their mean values
i, . and
write Dn
= D+dn,
Sn
= 5+Sn
and In = i+in •
5,
Sand
Then eqn.b) becomes:
= 5.(5n_1 /5 -
in-II 7: )
Around the respiratory frequency, the baroreflex-equation a) may be used in the
bit) d n
form:
in =
bill) d
n
ao.sn=
So eqn.blt) can be written as:
5.sn_I .(1/5 -
aol
r )
Hence, no diastolic variability exists (dn = 0) if a o =
S=
r: Is.
We consider here
the case of
i5
7.3.2) and
"t = 1850 ms, and we find: d n = -O.01.sn . The respiratory varia-
= 75 mmHg,
143 mmHg,
a o = 13.2 ms/mmHg (section
tions in D are only one per cent of the variations in S and no respiratory peak
is expected in the power spectrum of D.
Fig.3b illustrates qualitatively the argument presented above. A few consecutive heart beats are depicted. The systolic pressure is higher during beat n by
112
..
,-, ,
I
...
..
Fig.7-3b If the systolic value Sn is
higher than the previous value Sn-l and
if tre interval length In does not change,
trediastolic pressure Dn+l will be higher
than the previous one Dn (broken lines) .
However, the baroreflex transforms the
increased systolic pressure into a longer
interval and so the diastolic pressure
does not vary (solid line). This mechanism explains the lack of respiratory variabili ty in the diastolic pressures.
an amount .A S, due to respiratory influences (e.g. increased filling of the left
ventricle). The higher value of Sn would lead to an increased diastolic value
Dn+l if the interval length In were equal to the length of the previous beat
~l
(broken lines). However, the higher systolic pressure goes together with a
longer interbeat interval In (baroreflex ?!) and hence with a longer diastolic
run-off period, which tends to decrease the diastolic pressure (solid line). The
net effect is a more or less unchanged value of the diastolic pressure:
Dn+l ~ Dn'
Consequently, it can be argued that the lack of respiratory variability in the
diastolic pressures is a token of a functioning baroreflex (DeBoer et al.,
1985f,g). The argument presented above is only valid for the variability in the
respiratory frequency-band, because only there the systolic pressures and the in-tervals vary together. Therefore the 10-s variability in the diastolic pressures
is not suppressed (d.
fig.lb).
7.4.4 Relations between the spectra
As the pulse pressure P n is defined as the difference between systolic pressure
~
and diastolic pressure On' it is to be expected that the power spectrum of P
and the phase spectrum of P against I can be derived from the power and
phase spectra of Sand D. Figs.4a,b show the directly calculated power spectrum of P and phase spectrum of P against I (solid lines), and the spectra as
computed from the spectra of Sand D (dotted lines; only between 0.05 Hz
and 0.15 Hz).
For frequencies above 0.15 Hz, little variability in diastolic
113
pressure exists (fig.1b) and hence for these frequencies the spectra of P and of
P against I (figs.4a,b) resemble the spectra of S and of S against I (figs.1b,c),
respecti vel y.
In the range 0.05-0.15 Hz the spectrum of P was calculated from the spectra
of Sand D as indicated in fig.4c. The Fourier transform of P is a complex
......
quantity, having amplitude and phase. The components of P can be obtained by
.....
~
vector-subtraction of the Fourier components of Sand D.
The example in
.....
...
fig.4c shows the calculation for f = 0.1 Hz. The phase of I was defined as 00 •
~
....
~
The phase differences between S and I (-62 0 ) and between D and I (-79 0 ) were
....
.....
...
read from fig.lc and fig.1d, respectively. The lengths of the vectors Sand D
....
were taken from fig. lb. The vector P was found as the difference between S
100.0
P (f)
1.0
a
k 2 (f)
p-
\.
b
/-
P-I
I~
(mmHg) 21Hz
O.
50.0
I
I
I
I
I
180
J
r ,,,'\,,__
\
\
\
\
,
cP
(f)
90
0
-90
T
114
Fig.7-4a Power spectrum P(f) of pulse
pressure P, calculated directly (solid
line), and from the power and
cross-spectra of Sand D (dotted line;
only between 0.05 Hz and 0.15 Hz)_
Fig.7-4b Squared coherence spectrum
k2(f) (dashed) and directly calculated
phase spectrum cP (f) (solid line) of P
against L The dotted line is the phase
spectrum as calculated from the power
and cross-spectra of Sand D.
Fig.7-4c Diagram of the calculation of
the Fourier components of P for
f = 0.1 Hz by vector-subtraction of the
Fourier components of Sand D. Phase
angles and lengths of the vectors Sand
D are read from figs.l b-d.
...a
and 1); its (squared) length and phase angle (_21 0 ) are plotted in figs.4a,b (dotted lines). The correspondence between the dotted line and the directly calculated values is good, which was to be expected. This shows that the spectra of
D, S andP are not independent, and that each of them may be computed from
the other two.
7.5 Discussion
We presented a beat-to-beat model of part of the cardiovascular system. We
checked whether this model could be confirmed by the power and cross-spectra
of blood pressure and RR-intervals (chapter 6, or DeBoer et aI., 1985). The
baroreflex equation a), relating interval length to systolic pressure, held approximately muy for respiratory frequencies (0.20-0.35 Hz). The Windkessel equation b), describing the diastolic pressure as a function of previous systolic pressure and interval, was confirmed by the shape of the phase spectrum of L against I, with the logarithmic decrement L defined as Ln ::: log(Sn_l/Dn).
Using the model equations, it becomes clear why the respiratory variations in
the diastolic pressure values are small: a high value of the systolic pressure
would be followed by a high diastolic value; but variations in systolic pressure
and in interval go together at the respiratory frequency, possibly due to the
fast baroreflex, and so the high systolic value implies a lengthened interval,
which tends to lower the diastolic pressure. As these two influences are counteracting, the diastolic pressure will remain more or less unaffected under resplratoryvariatio~of
systo!icpressure andintervah Our model states that the
absence of respiratory influence in the diastolic pressure is consequential to a
functioning baroreflex, and so a diminished vagal regulation of heart rate should
become manifest in the appearance of a respiratory peak in the spectrum of
diastolic pressures.
Scher and Young (1970) altered the blood pressure in unanesthetized dogs by sinusoidal inflation of implanted aortic cuffs and found a good correlation
between interval duration and systolic pressure during the interval. They considered this "no-lag" response as vagal and found values of the baroreflex sensitivity coefficient between 10 and 45 ms/mmHg (ct. section 7.3.1). When the
pressure was considerably decreased, the interval-response was slower, due to
sympathetic effects. Using stimulation frequencies of 0.05-0.20 Hz, they found
115
little or no phase difference between variation in systolic pressure and interval.
This is not in line with our results, as we find 60 0 _90 0 phase difference around
0.1 Hz. Possibly the spontaneous fluctuations in heart rate and blood pressure
as studied by us are not directly comparable with the imposed pressure variations in the study of Scher and Young. The difference in species may also be
important; dogs usually show a much larger respiratory sinus arrhythmia than
is seen in man, which suggests a difference in the cardiovascular control system. In addition, the posture of dogs is often different from the posture of
men.
One is tempted to approximate the phase spectrum of systolic pressure against
interval (fig. Ie) by a line from -90 0 at f :: 0 Hz to +90 0 at f = 1/21 = 0.54 Hz.
This line corresponds with the phase spectra belonging to the difference equations ~ :: 0('(Sn-5n+2) or 5 n = (3·(In-1n-2)' as can be seen using Z-transforms
(Jury,1973; Jenkins and Watts, 1968, ch.8.4); the first of these equations is
then expressed by:
I(Z)::
(X'(5(Z>-Z25(Z»::
ex ·5(Z).(1-Z2),
and, if we put Z :: exp(21TifI):
I(Z) =
a· 5(Z).(l-exp(4lf' in).
This amounts to a phase difference between 5 and I that goes linearly from
_900 for f :: 0 to +90 0 for f = 1/21, i.e. similar to the phase spectrum of
fig.lc.
The
second
difference
equation,
when
expressed
as
S(Z) :: (3.I(Z).(l-Z-2) = r·I(Z).(l-exp(-41Tifl), leads to the same phase spectrum.
However, both difference equations are non-causal, as they predict an interval
from a subsequent systolic pressure, and a systolic pressure from a not yet
ended interval, respectively. Therefore they do not seem to be attractive for
modelling the physiological relationship between 5 and I. The difference equations are more fully discussed in appendix 7./\1.
Hence, we have no explanation for the shape of the phase spectrum of systolic
pressures 5 against intervals I. This spectrum is not in conformity with the
simple baroreflex influence as modelled by eqn.a). Both difference equations
which we present describe the shape of the phase spectrum, but a physiological
interpretation is not available. The lead of around 2 s of systolic pressure variabilityon interval variability in the 10-second region may be related to sympathetic regulation, which has latencies in this range (Levy and Martin, 1979).
116
It appears to be necessary to model the influence of blood pressure on
RR-intervallength in a more elaborate way than was done in this paper. On
physiological grounds the model should at least include some sympathetic regulation of heart-rate, in addition to the vagal regulation of equation a).
We showed that the spectrum of pulse pressures can be derived from the spectra of systolic and diastolic pressures, due to the identity P n = Sn-Dn (section
7.3.4). In a similar way other -- more interesting - relationships between pressures and/or intervals should be reflected in the various spectra.
Two final remarks should be made. First, the direct relationship in variability
of interval and of systolic pressure in the respiratory region need not be caused
solely by baroreflex influence, but may also be due to a simultaneous influence
of respiration-coupled efferent nervous activity on RR-interval and on blood
pressure, since correlation does not imply causality. The fact that for higher
respiratory frequencies the interval seems to lead the systolic pressure may also
be an indication for this. Second, it need not be that a single regulatory mechanism is commanding both the lO-second-variability and the respiration-linked
variability, and so different explanations may be needed for both phenomena.
117
7.AI Appendix l:
Phase spectra of the difference equations
In section 7.5 we argued that the phase spectra between interval and
systolk-pressure variability can be modelled by any of two different difference
equations, but we were not able to give a physiological interpretation of these
equations.
In this appendix the power and phase spectra belonging to these
difference equations are shown for simulated data. We also give two difference
equations that model the shape of the phase spectra of intervals and diastolic
pressures and again show calculated spectra of simulated data.
The phase spectra between intervals and systolic pressures resemble a straight
line between -90 0 at 0 Hz and +90 0 at
(fig.lc, 2c, 3c).
IIi Hz, with i the mean interval length
The following difference equations produce identical phase
spectra:
(AO,
and:
(A2).
Using Z-transforms, we found in section 7.5 that these equations can be written
as:
I(Z) =
ex· S(Z).(l-exp(4 Trifl»
= -2 0(·S(Z).sin(2 Trfi).i.exp(21T ifi)
(A!),
and:
S(Z) = ~. I(Z).(l-exp(-4 rr iff» = 2~' I(Z).5in(211" fI).i.exp(-2 'jj" ifi)
(A2),
respecti vel y.
Both equations describe an identical phase relation between systolic pressures
and intervals, but the amplitude relations between the interval spectrum and
the spectrum of systolic pressures as implied by equations Al and A2 are different. For eqn.AI the ratio between the power spectra of intervals PI(f) and
of systolic pressures PS(f) is PI(t)/Ps(t) = (1(Z)/S(Z»2 = 4 0( 2sin 2(2 nih for
eqn.A2 this ratio is 1/(4'~ 2sin2(21T'fi».
Hence, for f .... 0 Hz PS(f»> PI(f)
(eqn.A!), or PI(f)«
The same holds for f -+01/(21).
PS(f) (eqn.A2).
Similarly, difference equations can be constructed that model the phase spectrum of diastolic pressure against intervals; this spectrum has approximately
the value _90 0 over the whole frequency range (fig.ld). J)ifference equations
that produce a constant phase difference -90 0 are:
In = Dn_I-Dn+l
118
(/\3),
as well as:
On :: In+1-ln-1
(A4).
The phase spectra for these two difference equations are again found by means
of the Z-transform:
I{Z):: 0(Z)(Z-1_Z) :: 0{Z).(exp(-2lfifl)-exp(21TifI»
==
-20{Z).sin(21T'fi).i
(A3),
and:
Del)
==
I(Z)(Z-Z-l)
==
I(Z).(exp(2lrifl)-exp(-21Tifl» :: 2I(Z).sin(2lT'fi).i
(A4).
Eqns.A3 and A4 lead to similar amplitude relations between the spectra of intervals (I) and of diastolic pressures (D) as was the case for intervals and systolic pressures (5) as shown above.
Hence, both for the spectrum of I against 5 and of I against 0, two difference
equations are found to agree with the phase spectra. This dual set of difference equations is to be compared with the similar situation if differential equations are sought that may explain a phase-difference of 90 0 between two sig-'
nals x(t) and yet): the two differential equations x(t)
==
y,(t) and yet) :: -x~t)
imply as a relation between the Fourier-transforms X( IN)
yew)
==
==
i w·Y( w) and
-it.l·X(W), respectively, and both relations lead to a phase-difference of
90 0 (but different ratios of the power spectra).
The two difference equations mimicking the phase spectrum of I against Dare
not causally interpretable.
To illustrate the above considerations, we show in figs.AI,2 power and
crosS-spectra of simulated "systolic pressures". "diastolic pressures" and "intervals", using the difference equations AI-A4. For Fig.AI, the systolic values Sn.
diastolic values On and intervals In were computed for n
Sn
:: 100 +
In
=
Dn
Mn
with &n'
==
1 to 975 as:
On.
1 + 0.05 (Sn - Sn+2) +
50 + 20 (In+l - In_I) +
(d. eqn.A!),
(d. eqn.A4),
75 +
~ n'
En' 7n and ~ n independent Gaussian noise with mean zero and
standard deviation 5, 0.0.5, I and 4, respectively.
Note that the simulated
"mean values" Mn consist only of noise.
119
100.0
1.0
- 5 88.115
-11.00
0.38
4.&4
b
a
o.
50.
1.0
1000.0
IS
/
,
d
II
1
1
180
",.---------------',
,-./-'
90
'
O.~,~--------~~~----------H
o
-90
-180
0.5
0.0
0.0
180
1.0
90
0
180
,-/-------------'-,
/
,
,,
I
I
ID
,
\
\
\"\
I
O.
\
I
I
I
I
-90
I
1
I
J
0.0
0.0
Fig.7-Al Power spectra of I (a), S and M (b) and 0 (c), and
cross-spectra of S against I (d), M against I (e) and 0 against I (f).
The data are simulated by difference equations (see text). Most noteworthy are the shape of the phase spectrum of S against I, going
from _90° at f = 0 Hz to +90 0 at f = 0.5 Hz, and of the phase spectrum of 0 against I, at a constant value of -90°. These phase spectra
resemble the ones computed from experimental data. The spectra are
f uUy discussed in the text. Horizontal: frequency (Hz), vertical:
power (figs.Ala-c), coherence (figs.Ald-f, dashed) and phase (figs.Ald-f,
solid).
120
90
\
0
-90
1000.0
1.0
--11.00
0.35
- S 100.00
12.20
a
- - - -M 75.04
4.01
O.
500.0
1000.0
/---------------- .....,
1.0
-D
411.89
4.88
,""/
I
I
IS
c
I
I
180
\
d
\
\
I
500.0
0.5,~-----~~------r~
90
o
-90
0.0
0.0
1.0
0.1
0.2
0.3
0.4
0.5
180
IM
90
O.
0
-90
1.0
180
IO
90
o
0.5
r----------------------~ -90
o. 0,,-::--'---:-,-::--,0.0
0.1
Fig.7-A2 As Fig.Al, but a different set of difference equations is
used for the simulation.
121
For Fig.A2, Sn' Dn' In and Mn were computed as:
Dn = 50 +
&n'
In
Sn
= 1 + 0.05 (Dn_l - D n+1) +
tn'
= 100 + 20 (In - In_2) +
YLn>
Mn
= 75 +
with
hn'
(cf. eqn.A3),
(cf. eqn.A2),
~ n'
€ n'
~ nand
~ n as above.
The phase spectra between I and S (figs.Ald and A2d) are seen to be identical
and in correspondence with theory; likewise the phase spectra between I and 0
(figs.Alf and A2f). The power spectra are "white" for Sin fig.Alb, and for 0
in fig.A2c. The power spectra of I in fig.Ala and fig.A2a show the expected
sin2(2m)-shape: low values for f = 0 Hz and f = 1/21 = 0.5 Hz, and a maximum for f = 1/41 = 0.25 Hz.
The power spectra of 0 in fig.Ale and of Sin fig.A2b have a sin 4(211fI)-shape.
The power spectra of M are white (figs.Alb, A2b), and the coherence spectra
of I against M have low values throughout (figs.Ale, A2e).
In summary: this appendix shows that the stated difference equations lead to
results resembling the experimentally obtained phase spectra of I against Sand
of I against 0; however, the non-causal character of the equations hinders a
physiological interpretation of the equations.
122